animate Scaling part of an object driven by another object
Consider using a cast modifier.
Make sure you original mesh has many vertices.
Use the object field to determine the origin.
The object field can be an empty to control the origin.
When you move the empty the displacement will move also.
If your axis is World Z axis aligned you may choose to keep z unchanged in the modifier.
More to follow.
Consider also using a shrink wrap if you want more control over the modified shape.  Not yet shown.
Create slanted end at a pipe / cylinder by Rotation
I do not know your artistic vision for the deformation.  Here is a less smooth modification.